Represent linear relationships using graphs

A linear relationship is represented on the coordinate plane by ordered pairs $(x,y)$ , with the independent quantity on the horizontal axis and the dependent quantity on the vertical axis; its points lie on a straight line, including relationships with positive, negative, or zero rate of change. Graphs connect tables, equations such as $y=mx+b$ , and contexts by showing the meaning of the slope and intercept, without extending to nonlinear functions or more advanced forms.

Detailed Explanation: Represent linear relationships using graphs

A linear relationship can be shown by plotting its ordered pairs $(x,y)$ on a coordinate plane. The independent quantity, $x$ , goes on the horizontal axis, and the dependent quantity, $y$ , goes on the vertical axis. The points will form a straight line.

Example

Graph the relationship

y=2x+1.

Step 1: Make a table of values

Choose a few values for $x$ and calculate $y$ .

\begin{array}{c|c} x & y=2x+1\\ \hline -1 & 2(-1)+1=-1\\ 0 & 2(0)+1=1\\ 1 & 2(1)+1=3\\ 2 & 2(2)+1=5 \end{array}

This gives the ordered pairs

(-1,-1),\ (0,1),\ (1,3),\ (2,5).

Step 2: Label the axes

Draw a coordinate plane. Label the horizontal axis $x$ and the vertical axis $y$ . Use an even scale, such as counting by ones.

Step 3: Plot the points

Plot each ordered pair:

$(-1,-1)$ : move left $1$ and down $1$ .
$(0,1)$ : move up to $1$ on the $y$ -axis.
$(1,3)$ : move right $1$ and up to $3$ .
$(2,5)$ : move right $2$ and up to $5$ .

Step 4: Draw the line

Use a ruler to draw a straight line through the points. The line represents all the ordered pairs that follow the relationship $y=2x+1$ .

The number $2$ is the slope, so $y$ increases by $2$ whenever $x$ increases by $1$ . The number $1$ is the $y$ -intercept, so the line crosses the $y$ -axis at $(0,1)$ .

To check a graph of a linear relationship, make sure the points follow the equation and line up in a straight line.

Learn by doing: Represent linear relationships using graphs

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Slope - Find Equivalent - Slope Y Intercept Form to Graph

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